Calculus Examples

$lim_{x \to - 8} \frac{\sqrt{3 x^{4} + x}}{x^{2} - 8}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$\frac{lim_{x \to - 8} \sqrt{3 x^{4} + x}}{lim_{x \to - 8} x^{2} - 8}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 8} 3 x^{4} + x}}{lim_{x \to - 8} x^{2} - 8}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\frac{\sqrt{lim_{x \to - 8} 3 x^{4} + lim_{x \to - 8} x}}{lim_{x \to - 8} x^{2} - 8}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{3 lim_{x \to - 8} x^{4} + lim_{x \to - 8} x}}{lim_{x \to - 8} x^{2} - 8}$

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{3 {(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} x}}{lim_{x \to - 8} x^{2} - 8}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\frac{\sqrt{3 {(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} x}}{lim_{x \to - 8} x^{2} - lim_{x \to - 8} 8}$

Step 7

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{3 {(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} x}}{{(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 8}$

Step 8

Evaluate the limit of $8$ which is constant as $x$ approaches $- 8$ .

$\frac{\sqrt{3 {(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} x}}{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 8}$

Step 9

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt{3 {(- 8)}^{4} + lim_{x \to - 8} x}}{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 8}$

Step 9.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt{3 {(- 8)}^{4} - 8}}{{(lim_{x \to - 8} x)}^{2} - 1 \cdot 8}$

Step 9.3

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt{3 {(- 8)}^{4} - 8}}{{(- 8)}^{2} - 1 \cdot 8}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 8$ to the power of $4$ .

$\frac{\sqrt{3 \cdot 4096 - 8}}{{(- 8)}^{2} - 1 \cdot 8}$

Step 10.1.2

Multiply $3$ by $4096$ .

$\frac{\sqrt{12288 - 8}}{{(- 8)}^{2} - 1 \cdot 8}$

Step 10.1.3

Subtract $8$ from $12288$ .

$\frac{\sqrt{12280}}{{(- 8)}^{2} - 1 \cdot 8}$

Step 10.1.4

Rewrite $12280$ as $2^{2} \cdot 3070$ .

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Step 10.1.4.1

Factor $4$ out of $12280$ .

$\frac{\sqrt{4 (3070)}}{{(- 8)}^{2} - 1 \cdot 8}$

Step 10.1.4.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 3070}}{{(- 8)}^{2} - 1 \cdot 8}$

Step 10.1.5

Pull terms out from under the radical.

$\frac{2 \sqrt{3070}}{{(- 8)}^{2} - 1 \cdot 8}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Raise $- 8$ to the power of $2$ .

$\frac{2 \sqrt{3070}}{64 - 1 \cdot 8}$

Step 10.2.2

Multiply $- 1$ by $8$ .

$\frac{2 \sqrt{3070}}{64 - 8}$

Step 10.2.3

Subtract $8$ from $64$ .

$\frac{2 \sqrt{3070}}{56}$

Step 10.3

Cancel the common factor of $2$ and $56$ .

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Step 10.3.1

Factor $2$ out of $2 \sqrt{3070}$ .

$\frac{2 (\sqrt{3070})}{56}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $2$ out of $56$ .

$\frac{2 \sqrt{3070}}{2 \cdot 28}$

Step 10.3.2.2

Cancel the common factor.

$\frac{2 \sqrt{3070}}{2 \cdot 28}$

Step 10.3.2.3

Rewrite the expression.

$\frac{\sqrt{3070}}{28}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3070}}{28}$

Decimal Form:

$1.97884216 \dots$